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doi: 10.3934/dcds.2020351

A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity

A. Alexandrou Himonas 1,, and  Gerson Petronilho 2,

1. 

University of Notre Dame, Department of Mathematics, Notre Dame, IN 46556, USA
2. 

Universidade Federal de São Carlos, Departamento de Matemática, São Carlos, SP 13565-905, Brazil

* Corresponding author

Received  May 2020 Published  October 2020

The Cauchy problem of the modified Camassa-Holm (mCH) equation with initial data $ u(0) $ that are analytic on the line and have uniform radius of analyticity $ r(0) $ is considered. First, by using bilinear estimates for the nonlocal nonlinearity in analytic Bourgain spaces, it is shown that this equation is well-posed in analytic Gevrey spaces $ G^{\delta, s} $, with useful solution lifespan $ T_0 $ and size estimates. This shows that the radius of spatial analyticity $ r(t) $ persists during the time interval $ [-T_0, T_0] $. Then, exploiting the fact that solutions to this equation conserve the $ H^1 $ norm, and utilizing the available bilinear estimates, an almost conservation low in $ G^{\delta,1} $ spaces is proved. Finally, using this almost conservation law it is shown that the solution $ u(t) $ exists for all time $ t $ and a lower bound for the radius of spatial analyticity is provided.

Keywords: Modified Camassa-Holm equation, Cauchy problem, analytic spaces, uniform radius of analyticity, bilinear estimates, algebraic decrease, approximate conservation law.
Mathematics Subject Classification: Primary: 35Q53.
Citation: A. Alexandrou Himonas, Gerson Petronilho. A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020351
References:
[1]

R. F. Barostichi, A. A. Himonas and G. Petronilho, Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems, J. Funct. Anal., 270 (2016), 330-358.  doi: 10.1016/j.jfa.2015.06.008.  Google Scholar

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J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.  doi: 10.1016/j.anihpc.2004.12.004.  Google Scholar

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J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 2: KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 1: Schrödinger equation, Geom. Funct. Anal., 3 (1993), 209-262.   Google Scholar

[5]

J. Bourgain, On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl., 1993 (1995), 17-86.   Google Scholar

[6]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[7]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[10]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.   Google Scholar

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperi-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[13]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[14]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.   Google Scholar

[15]

R. Figuera, A. A. Himonas and F. Yan, A higher dispersion KdV equation on the line, Nonlinear Anal., 199 (2000), 112055, 38 pp. doi: 10.1016/j.na.2020.112055.  Google Scholar

[16]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[17]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/1982), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[18]

Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations, 15 (2002), 1325-1334.   Google Scholar

[19]

A. A. Himonas, H. Kalisch and S. Selberg, On persistence of spatial analyticity for the dispersion-generalized periodic KdV equation, Nonlinear Anal. Real World Appl, 38 (2017), 35-48.  doi: 10.1016/j.nonrwa.2017.04.003.  Google Scholar

[20]

A. A. Himonas and G. Misiołek, Global well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161 (2000), 479-495.  doi: 10.1006/jdeq.1999.3695.  Google Scholar

[21]

A. A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations, 22 (2009), 201-224.   Google Scholar

[22]

A. A. Himonas and G. Misiołek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23 (1998), 123-139.  doi: 10.1080/03605309808821340.  Google Scholar

[23]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584.  doi: 10.1007/s00208-003-0466-1.  Google Scholar

[24]

H. Hirayama, Local well-posedness for the periodic higher order KdV type equations, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 677-693.  doi: 10.1007/s00030-011-0147-9.  Google Scholar

[25]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93-128.   Google Scholar

[26]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.  doi: 10.1016/S0294-1449(16)30377-8.  Google Scholar

[27]

Y. Katznelson, An Introduction to Harmonic Analysis Corrected ed., Dover Publications, Inc., New York, 1976.  Google Scholar

[28]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[29]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[30]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principl, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[31]

C. E. Kenig, G. Ponce and L. Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 122 (1994), 157-166.  doi: 10.1090/S0002-9939-1994-1195480-8.  Google Scholar

[32]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[33]

J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217 (2005), 393-430.  doi: 10.1016/j.jde.2004.09.007.  Google Scholar

[34]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[35]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext Springer, New York, 2009.  Google Scholar

[36]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[37]

S. Selberg and D. O. da Silva, Lower Bounds on the radius of a spatial analyticity for the KdV equation, Ann. Henri Poincaré, 18 (2017), 1009-1023.  doi: 10.1007/s00023-016-0498-1.  Google Scholar

[38]

T. Tao, Nonlinear Dispersive Equations-Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

show all references

References:
[1]

R. F. Barostichi, A. A. Himonas and G. Petronilho, Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems, J. Funct. Anal., 270 (2016), 330-358.  doi: 10.1016/j.jfa.2015.06.008.  Google Scholar

[2]

J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.  doi: 10.1016/j.anihpc.2004.12.004.  Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 2: KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 1: Schrödinger equation, Geom. Funct. Anal., 3 (1993), 209-262.   Google Scholar

[5]

J. Bourgain, On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl., 1993 (1995), 17-86.   Google Scholar

[6]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[7]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[10]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.   Google Scholar

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperi-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[13]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[14]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.   Google Scholar

[15]

R. Figuera, A. A. Himonas and F. Yan, A higher dispersion KdV equation on the line, Nonlinear Anal., 199 (2000), 112055, 38 pp. doi: 10.1016/j.na.2020.112055.  Google Scholar

[16]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[17]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/1982), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[18]

Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations, 15 (2002), 1325-1334.   Google Scholar

[19]

A. A. Himonas, H. Kalisch and S. Selberg, On persistence of spatial analyticity for the dispersion-generalized periodic KdV equation, Nonlinear Anal. Real World Appl, 38 (2017), 35-48.  doi: 10.1016/j.nonrwa.2017.04.003.  Google Scholar

[20]

A. A. Himonas and G. Misiołek, Global well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161 (2000), 479-495.  doi: 10.1006/jdeq.1999.3695.  Google Scholar

[21]

A. A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations, 22 (2009), 201-224.   Google Scholar

[22]

A. A. Himonas and G. Misiołek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23 (1998), 123-139.  doi: 10.1080/03605309808821340.  Google Scholar

[23]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584.  doi: 10.1007/s00208-003-0466-1.  Google Scholar

[24]

H. Hirayama, Local well-posedness for the periodic higher order KdV type equations, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 677-693.  doi: 10.1007/s00030-011-0147-9.  Google Scholar

[25]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93-128.   Google Scholar

[26]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.  doi: 10.1016/S0294-1449(16)30377-8.  Google Scholar

[27]

Y. Katznelson, An Introduction to Harmonic Analysis Corrected ed., Dover Publications, Inc., New York, 1976.  Google Scholar

[28]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[29]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[30]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principl, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[31]

C. E. Kenig, G. Ponce and L. Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 122 (1994), 157-166.  doi: 10.1090/S0002-9939-1994-1195480-8.  Google Scholar

[32]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[33]

J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217 (2005), 393-430.  doi: 10.1016/j.jde.2004.09.007.  Google Scholar

[34]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[35]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext Springer, New York, 2009.  Google Scholar

[36]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[37]

S. Selberg and D. O. da Silva, Lower Bounds on the radius of a spatial analyticity for the KdV equation, Ann. Henri Poincaré, 18 (2017), 1009-1023.  doi: 10.1007/s00023-016-0498-1.  Google Scholar

[38]

T. Tao, Nonlinear Dispersive Equations-Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

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